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The theory of dual coding holds that the formation of mental images, in tandem with verbal processing, is often very helpful for learners. In other words, if we support verbal reasoning with visual representations, then better learning happens.

Many years ago, I was taught the dual coding technique outlined below to help with SUVAT problems. Of course, it wasn’t referred to as “dual coding” back then, but dual coding it most definitely is.

I found it a very useful technique at the time and I still find it useful to this day. And what is more, it is in my opinion a pedagogically powerful procedure. I genuinely believe that this technique helps students understand the complexities and nuances of SUVAT because it brings many things which are usually implicit out into the open and makes them explicit.

SUVAT: “Made darker by definition”?

BOSWELL. ‘He says plain things in a formal and abstract way, to be sure: but his method is good: for to have clear notions upon any subject, we must have recourse to analytick arrangement.’

JOHNSON. ‘Sir, it is what every body does, whether they will or no. But sometimes things may be made darker by definition. I see a cow, I define her, Animal quadrupes ruminans cornutum. But a goat ruminates, and a cow may have no horns. Cow is plainer.

— Boswell’s Life of Johnson (1791)

As I see it, the enduring difficulty with SUVAT problems is that such things can indeed be made darker by definition. Students are usually more than willing to accept the formal definitions of s, u, v, a and t and can apply them to straightforward and predictable problems. However, the robotic death-by-algorithm approach fails all too frequently when faced with even minor variations on a theme.

Worse still, students often treat acceleration, displacement and velocity as nearly-synonymous interchangeable quantities: they are all lumped together in that naive “intuitive physics” category called MOVEMENT.

The approach that follows attempts to make students plainly see differences between the SUVAT quantities and, hopefully, as make them as plain as a cow (to borrow Dr Johnson’s colourful phrasing).

Visual Symbols for the Dual-coding of SUVAT problems

1.1 Analysing a simple SUVAT problem using dual coding

Problem: a motorcycle accelerates from rest at 0.8 m/s2 for a time of 6.0 seconds. Calculate (a) the distance travelled; and (b) the final velocity.

Please note:

We are using the AQA-friendly convention of substituting values before rearrangement. (Some AQA mark schemes award a mark for the correct substitution of values into an expression; however, the mark will not be awarded if the expression is incorrectly rearranged. Weaker students are strongly encouraged to substitute before rearrangement, and this is what I model.)

A later time is indicated by the movement of the hands on the clock.

So far, so blindingly obvious, some might say.

But I hope the following examples will indicate the versatility of the approach.

Problem: A coin is dropped from rest takes 0.84 s to fall a distance of 3.5 m so that it strikes the water at the bottom of a well. With what speed must it be thrown vertically so that it takes exactly 1.5 s to hit the surface of the water?

Another advantage of this method is that it makes assigning positive and negative directions to the SUVAT vectors easy as it becomes a matter of simply comparing the directions of each vector quantity (that is to say, s, u, v and a) with the arbitrarily selected positive direction arrow when we substitute values into the expression.

But what would happen if we’d selected a different positive direction arrow?

Problem: A well is 3.5 m deep so that a coin dropped from rest takes 0.84 s to strike the surface of the water. With what speed must it be thrown so that it takes exactly 1.5 s to hit the surface of the water?

The answer is, of course, numerically equal to the previous answer. However, following the arbitrarily selected down is positive convention, we have a negative answer.

1.3 Analysing a projectile problem using dual coding

Let’s look at this typical problem from AQA.

We could annotate the diagram like this:

Guiding our students through the calculation:

Just Show ‘Em!

Some trad-inclined teachers have embraced the motto: Just tell ’em!

It’s a good motto, to which dual coding can add the welcome corollary: Just show ’em!

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The famous phrase is, of course, from physicist Eugene Wigner (1960: 2):

My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.

Further exploration of the above problem using dual coding can, I believe, give A-level students a glimpse of the truth of Wigner’s phrase.

This Is The Root You’re Looking For

In the calculation above, we found that when s = -1.8 m, v could have a value of plus or minus 6.90 m/s. Since we were interested in the velocity of the kite boarder at the end of the journey, we concluded that it was the negative root that was significant for our purposes.

But does the positive root have any physical significance? Why yes, it does. It indicates the other possible value of v when s = -1.8 m.

The displacement was -1.8 m at only one point on the real journey. However, if the kite boarder had started their projectile motion from the level of the water surface instead of from the top of the ramp, their vertical velocity at this point would have been +6.9 m/s.

The fact that the kite boarder did not start their journey from this point is immaterial. Applying the mathematics not only tells us about their actual journey, but all other possible journeys that are consistent with the stated parameters and the subset of the laws of physics that we are considering in this problem — and that, to me, borders enough on the mysterious to bring home Wigner’s point.

And finally…

This information allows us to annotate our final diagram as below (bearing in mind, of course, that the real journey of the kite boarder started from the top of the ramp and not from the water’s surface as shown).

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

I am very pleased to say that I got my best annual viewing figure ever: just over 9000 views in total beating a previous best of 7000.

Small beer for some in the edu-blogosphere perhaps, but I am well chuffed.

And my most popular blogs were (in reverse order):

5) IoP Energy for Busy Teachers. This is yet another of my periodic tilts at the Institute of Physics’ revised schema for teaching energy, including some attempted humour.

4) The FBI and Gang Sign for Physicists. I am am at a loss to explain why this random stream-of-edu-conciousness post from 2016 seems to get a slow but steady stream of readers, mainly from the U.S.A. for some reason

3) Teaching Electric Circuits? Climb On Board The Coulomb Train! This, I have to admit, is one of my personal favourites. Although a persuasive case can be made for the rope model (I’m looking at you, @PhysicsUK and this), the CTM (Coulomb Train Model) is still the best IMHO. For example, which one would be the best when we’re considering RMS values, huh?

2) Two posts on applying the Singapore Bar Model to GCSE Science Topics and ditching those horrible, horrible formula triangles. The first was on Magnification and the second on Kinetic Energy.

This is my contribution to the #CurriculuminScience symposium. You can read the first contribution from Ruth Walker here. The next contribution from Jasper Green can be found here.

“She said she was going to join a church as soon as she decided which one was right. She never did decide. She did develop a terrific hankering for a crucifix, though. And she bought one from a Santa Fe gift shop during a trip the little family made out West during the Great Depression. Like so many Americans, she was trying to construct a life that made sense from things she found in gift shops.”

— Kurt Vonnegut, Slaughterhouse-Five [emphasis added]

Introduction

It was never supposed to be like this, of course. Many of the great thinkers of the past conceived of the human mind as a vast pyramid: either an inverted pyramid resting on an apex consisting of a single, unfalsifiable thought such as “I think therefore I am” as Rationalists such as Descartes posited; or, alternatively, as a pyramid resting on a base of simple sense-impressions as Empiricists such as Locke suggested.

The examples of entities that exist are taken from Douglas Adams’ The Hitchhiker’s Guide To The Galaxy, where the sentient computer Deep Thought started with “I think therefore I am” and deduced the existence of rice pudding and income tax before anyone managed to switch it off.

The truths emerging from modern cognitive science indicate that things are a good deal more complicated and messier than either the Rationalists or Empiricists supposed.

In fact, all of us are closer to Mrs Pilgrim in Vonnegut’s Slaughterhouse-Five than we would generally like to admit: the uncomfortable truth is that we are all closer to opportunistic concept-grubbing, “gift shop”-magpies than the systematic pyramid-masons of either Rationalist or Empiricist thought. Each and every one of us is, to a greater or lesser degree, “trying to construct a life that makes sense” from random things that we find lying around in real or metaphorical gift shops.

Perhaps (of all people!) Dashiell Hammett put it best:

“Nobody thinks clearly, no matter what they pretend. Thinking’s a dizzy business, a matter of catching as many of those foggy glimpses as you can and fitting them together the best you can. That’s why people hang on so tight to their beliefs and opinions; because, compared to the haphazard way in which they arrived at, even the goofiest opinion seems wonderfully clear, sane, and self-evident. And if you let it get away from you, then you’ve got to dive back into that foggy muddle to wangle yourself out another to take its place.”
— Dashiell Hammett, The Dain Curse

Defeat From The Jaws of Victory: “Here’s to you, Mrs Pilgrim.”

Andrea diSessa (1996) recounts a series of interviews with “J”, a freshman undergraduate student of Physics at university. During one interview, J was asked to explain the physics of throwing a ball up into the air. She recounted a near-perfect, professional physicist-level analysis of the phenomenon, noting (correctly) that after the ball leaves the hand the “only force acting on it is gravity”. However, when diSessa asked the seemingly innocuous question about what happens at the peak of the toss:

Rather than produce a straightforward answer, J proceeded to reformulate her description of the toss. The reformulation is not instantaneous . . . Strikingly she winds up with an “impetus theory” account of the toss. “Your hand imparts a force that at first overcomes gravity, but gradually dies away. At the peak, there a balance of forces, which is broken as the internal force fades further and gravity takes over.”

In other words, even a student of Physics, educated to a much higher level of domain-specific knowledge than the typical layperson, can be persuaded to retreat back into the ”foggy muddle” with surprising ease. In other words, even the very best of us can snatch defeat from the jaws of victory all too easily.

diSessa (1988) explains this and similar models as part of the KIP model (Knowledge in Pieces). For example:

intuitive physics is a fragmented collection of ideas, loosely connected and reinforcing, having none of the commitment or systematicity that one attributes to theories.

The basic “atom” or building block of this empirical model is the p-prim or phenomenological primitive.

P-prims are elements of intuitive knowledge that constitute people’s “sense of mechanism”, their sense of which happenings are obvious, which are plausible, which are implausible, and how one can explain or refute real or imagined possibilities. [diSessa 2018: 69]

P-prims are abstractions of familiar events that come to serve as explanations as they are applied to a wider range of contexts. The p indicates that they originate from the phenomenologically-rich and lived experience of human beings; the prim indicates that they are primitive in the sense that they sufficient explanations of phenomena. Once a p-prim is invoked, usually no further explanation is required or possible: “That’s just the way it is.” Examples of p-prims suggested by diSessa [1996: 716] are:

The “Ohm’s Law” p-prim: the idea that an outcome increases as a “force” increases, but decreases as the “resistance” increases.

The “Balance and Equilibrium” p-prim: systems which are “in balance” will be stable; systems which are “out of balance” will naturally and spontaneously return to equilibrium.

The “Blocking and Guiding” p-prim: solid and stable objects can stop objects moving without applying a force; tubes and railway tracks can also “guide” moving objects without applying any force.

The “Dying Away” p-prim: lack of motion or activity is the natural state of inanimate objects; if disturbed, they will naturally return to this state as the perturbation “dies away’

P-prims are subconceptual: they comprise a fluid and changeable layer below concepts and beliefs. Humans may have hundreds if not thousands of p-prims. There is no strict hierarchy: we may shift from one p-prim to another with simply a shift of attention. Where multiple p-prims conflict, one facet of the situation may cue the application of a particular p-prim rather than another. [see diSessa 1996: 715]

The Wrath of Kuhn: “So You Say You Want a Revolution?”

In his hugely-influential The Structure of Scientific Revolutions (1970), Thomas Kuhn suggested that scientific progress had two distinct phases:

Normal Science, where essentially scientists engaged in puzzle-solving activity but where the guiding paradigm or disciplinary matrix of the science is more or less accepted without question. An example might be pre-Copernican astronomy where astronomers made observations and predictions without questioning the geocentric model of the Solar System;

Revolutionary Science, where scientists realise their previously-successful paradigm is no longer able to adequately explain observed phenomena. An example might be the rejection of the Newtonian paradigm and the acceptance of Einsteinian relativistic physics in the early 1900s.

Scientific progress was thus viewed as a gestalt switch between two incommensurable systems of knowledge. One either sees a “Newtonian”-duck, or a “Relativistic”-rabbit. One cannot see both simultaneously.

One can either see a duck or a rabbit: but not both at the same time.

Kuhn’s work was immensely influential (perhaps overly influential) in a number of spheres; in the context of education, the heady seductiveness of Kuhn’s approach directly influenced what diSessa [2014: 5] dubs the “misconceptions movement”.

Broadly speaking, proponents thought that students had deeply entrenched but false beliefs. The solution seemed obvious: these false beliefs were barriers to learning that had to be rooted out and overcome (c.f. the Ohm’s Law p-prim above!) . Students had to be persuaded to ditch their false beliefs and accept the correct ones.

But what was the nature of these false beliefs? diSessa [2014:7] argues that some like Carey (1985) drew explicit parallels with Kuhn’s work, arguing that children undergo a paradigm-shift at about 10-years-old when they recognise that inanimate objects do not have intentions and begin to think of “alive” as describing a set of mechanistic processes. Others (argues diSessa) like McCloskey (1983) supposed that students begin school physics with a well-formed, coherent and articulate theory (with parallels to early medieval scientists such as Buridan and Galileo) that directly competes with and interferes with their acceptance of Newtonian physics.

However, all of these approaches can be categorised as being part of the “Misconceptions Movement”.

A positive influence of misconceptions studies was bringing the importance of educational research into practical instructional circles. Teachers saw vivid examples of students responding to apparently simple conceptual questions in incorrect ways. Poor performance in response to basic questions, often years into instruction, could not be dismissed.

[diSessa 2014: 6]

Another hugely positive influence of Misconceptions research was that it showed that students were not “blank slates” and that prior knowledge had a strong influence on future learning.

However, according to diSessa the misconceptions movement also had some pernicious negative influences:

It emphasised the negative contributions of prior knowledge: it almost exclusively characterised prior knowledge as either false or unhelpful which led to “conflict” models of instruction. Ironically, the explicit detailing of “wrong” ideas in order to “overcome” them led to them being strengthened for some students.

How learning was possible was not a matter that was often discussed in detail. The depth, coherence or strength of particular misconceptions was not always assessed: were they simply isolated beliefs or coherent theories of a similar nature to those held by working scientists? As a result, practical guidance on how to teach particular concepts was not always forthcoming.

Tourist: “Is This Way To Amarillo?” Local: “Well, I wouldn’t start from here if I were you.”

As a working Physics teacher, one of the most useful teaching tools that I’ve begun using as a result of becoming aware of diSessa’s work, is that of a bridging analogy. This approach was outlined by Hammer 2000: S54-55. For example, how can we successfully introduce the idea of a normal reaction force, say in the context of a book resting on the surface of a table?

Students often invoke the “blocking” p-prim in this context. The table passively “blocks” the action of gravity — and that’s all there is to it.

However, a bridging analogy can be used here. Show an object resting on (and compressing) a spring; identify the forces acting on the object. Because the spring is an “active” component in this situation, students can accept that pushing down on it produces an upward “reaction force”. One can then extend this to (say) a student sitting on a plank (which “bows” slightly with their weight) and then apply it to more stable structure such as a table which exhibits no visible “bowing”.

I have found such approaches to be the most productive: in other words, we aim to work around the p-prim rather than attacking the p-prim head on, and along the way we try to get our students to activate more helpful p-prims that have more direct applicability to the context.

As teachers, we only very rarely have the luxury of choosing our students’ starting points. There is no “Well, if you want to get where you’re going, I wouldn’t start from here if I were you.”

We are teachers. Whatever the situation, we start from where our students start. Ladies and gentlemen, we start from here.

diSessa, A. A. (2018). A Friendly Introduction to “Knowledge in Pieces”: Modeling Types of Knowledge and Their Roles in Learning. In Invited Lectures from the 13th International Congress on Mathematical Education (pp. 65-84). Springer International Publishing. [Accessed from https://link.springer.com/chapter/10.1007/978-3-319-72170-5_5 on 22/10/18]

I’ve said it before and I’ll say it again: teaching electric circuits is hard.

Providing your students with a conceptual model can, in my opinion, be immensely helpful. In recent years, I have used what I call the Coulomb Train Model (CTM). It is essentially a variation on the “stiff chain” analogies that some researchers have argued as being particularly useful in developing students’ understanding.

One reason why I like the CTM is that it provides a physical picture to aid students’ comprehension of many of the electrical equations needed at GCSE.

Of course, any analogy or model will have its flaws, but on the whole I think the CTM has fewer than many of its rivals!

Essentially, the CTM pictures an electric circuit as a continuously moving chain of postively-charged “trucks” called coulombs that carry energy from the cell to (say) the bulb. In the diagram below, they should be pictured as moving clockwise.

The coulomb is, of course, the S.I. unit of electric charge, so rest assured that there is method in the apparent madness of naming our “trucks” with a word that would be unfamiliar to most of our students.

Charge flow = current x time

Charge flow = number of coulombs that pass a given point in time.

Current = number of coulombs that pass by in one second (i.e. current = charge flow / time).

In other words, an ammeter counts the coulombs passing by in one second. The ammeter only “sees” the coulombs and does not register how much (or how little) energy each one contains. Therefore current I1 and current I2 are equal.

The ammeters are shown as being semi-transparent in order to provide a visual cue that they are low resistance devices.

Energy transferred = charge flow x potential difference

On the CTM, potential difference can be pictured as energy being added to, or removed from, each coulomb.

For example, if one joule is removed from each coulomb as they pass through the bulb, the potential difference across the bulb is one volt. If one joule is added to each coulomb as they pass through the cell, then the potential difference (or e.m.f.) across the cell is one volt.

Amongst the myriad inconveniences and troubles of a Physics teacher’s life, the choice of the symbols commonly used to represent voltage, current and resistance, must surely rank in the top ten.

V is for voltage in volts, V

Well, OK, that’s sensible enough. On a good day, I may even remember to call it “potential difference”. The sage advice of Never use two words when one will do is commonly accepted by everyone; however, Physics teachers have, as a profession, decided to go it alone and completely ignore this tired old saw. Thus, voltage is become potential difference because of — erm, reasons (?)

One can only hope that everyone got the memo . . .

R is for resistance in ohms, Ω

R for resistance? That’s fairly sensible too.

“But what’s that weird squiggly thing, Sir?”

“Ah, you mean the Greek letter omega? Because Physics is soooo enormous that the measly 26 letters of the Latin alphabet ain’t big enough for it…”

I is for current in amps, A

“WTφ? Are you taking the πΣΣ, Sir?”

“I know, I know! Look, if it helps, think of it as short for intensité du courant . . . Wait, don’t leave! Stop, I have many more fun Physics facts to teach you! Look, here’s a picture of Richard Feynman playing his bongo drums — nooooooooo!”

Ohm’s Law: or is it more a sort of guideline?

V = I R

R = V / I

There is not a single instance where it is not true by definition. The value of resistance will always be equal to the ratio of the potential difference and the current.

Think of it like this. At room temperature, 1 V of potential difference can push (say) 0.5 A of current through the wire in a filament bulb. (I just love that retro 1890s tech, don’t you?)

This means it has a resistance of 1/0.5 = 2 ohms. However, bump up the potential difference to 6 V and the current is (say) 0.75 A. This means that is has a resistance of 6/0.75 = 8 ohms. Its resistance has changed because it has become hotter. In other words, its resistance is not constant.

Ohm’s Law is perhaps most simply stated as:

The potential difference is directly proportional to the current over a range of physical conditions (including temperature).

Using standard symbols:

V α I

or, taking R’ as a constant of proportionality:

V = I R’

You do see the difference, don’t you?

In the first example, R is not a constant value for a given range of physical conditions: for example it can get higher as the temperature increases.

In the second, R’ is constant over a range of temperatures and other physical conditions.

And so there we have it: V=IR can be a perfectly valid statement of Ohm’s Law, provided it is specified that R is constant. If one does not do that, then all bets are off…

In the meantime, here’s another picture of Richard Feynman playing the bongo drums. Enjoy!

@_youhadonejob tweeted this textbook picture with the amusing and sardonic comment “Little girl in this textbook is 5 m tall”.

I liked @jim_henderson60’s take on this when he tweeted: “You see. Physics helps us all grow tall.”

But then I started thinking, what if the 5 m measuring stick was in an inertial frame moving past Lottie’s inertial frame at a substantial fraction of light speed? (In my head, I named the girl “Lottie”, although “Alice” would be more in the more usual tradition of SR* pedagogy, I guess.)

The illustration could represent that single instant at which both ends of the 5 m ruler were precisely opposite Lottie’s head and feet as its inertial frame whizzed by hers…

A quick calculation indicated that Lorentz length contraction could indeed account for the relative measurements on the illustration if v = 0.97c

Of course, Lorentz length contraction is a two way street. From the 5 m ruler’s inertial frame, length contraction would make Lottie appear even shorter than her compact 1.2 m. Given that v = 0.97c, I calculate that she would appear only 0.29 m tall.

Correction: not appear. She would genuinely be only 0.29 m tall when viewed from that inertial frame, just as the 5 m rule would genuinely be only 1.2 m long when viewed from Lottie’s inertial frame.

We live in an universe where everything is indeed relative. However, for most of us that takes a fair amount of getting used to…

*SR = special relativity. My brain is currently too small to handle GR (general relativity).

AHT VAL: And once you’ve finished marking your students’ books and they have responded IN DETAIL to your DETAILED comments, you must take them in again and mark them a second time using a different coloured pen!

AHT HARVEY: A page that’s marked in only one colour is a useless page!

NQT BENJAMIN: Erm, if you say so. But why?

AHT VAL: It’s basic Ofsted-readiness, Benjamin. Without a clearly colour-coded dialogue between teacher and student, how can we prove that the student has made progress as a result of teacher feedback?

NQT BENJAMIN: But I’ve only got this red biro…

AHT HARVEY GRINS UNPLEASANTLY AND OPENS A CABINET FULL OF PENS OF MANY COLOURS.

AHT HARVEY: In this school we wage a constant battle against teacher sloth and indifference!

(With apologies to The League Of Gentlemen)

I have been a teacher for more than 26 years and I tell you this: I have never marked as much or as often as I am now. We are in the throes of a Marking Apocalypse — a Markopalypse, if you will.

And why am I doing this? Have I had a Damascene-road conversion to the joy of rigorous triple marking?

No. I do it because I have to. I do it because of my school’s marking policy. More to the point, I do it because my school expends a great deal of time and energy checking that their staff is following the policy. And my school is not unique in this.

Actually, to be fair, I think my current school has the most nearly-sensible policy of the three schools I have worked in most recently, but it is still an onerous burden even for an experienced teacher who can take a number of time-saving short cuts in terms of lesson planning and preparation.

Many schools now include so-called “deep marking” or “triple marking” in their lists of “non-negotiables”, but there are at least two things that I think all teachers should know about these policies.

[I]nspectors should not report on marking practice, or make judgements on it, other than whether it follows the school’s assessment policy. Inspectors will also not seek to attribute the degree of progress that pupils have made to marking that they might consider to be either effective or ineffective. Finally, inspectors will not make recommendations for improvement that involve marking, other than when the school’s marking/assessment policy is not being followed by a substantial proportion of teachers; this will then be an issue for the leadership and management to resolve.

2. “Students benefit from regular feedback”

Why yes, of course they do. But “feedback” does not necessarily equate to marking.

[F]eedback is conceptualized as information provided by an agent (e.g., teacher, peer, book, parent, self, experience) regarding aspects of one’s performance or understanding. A teacher or parent can provide corrective information, a peer can provide an alternative strategy, a book can provide information to clarify ideas, a parent can provide encouragement, and a learner can look up the answer to evaluate the correctness of a response. Feedback thus is a “consequence” of performance.

So a textbook, mark scheme or model answer can provide feedback. It does not have to be a paragraph written by the teacher and individualised for each student.

Daisy Christodoulo makes what I think is a telling point about the “typical” feedback paragraphs encouraged by many school policies:

[T]eachers end up writing out whole paragraphs at the end of a pupils’ piece of work: ‘Well done: you’ve displayed an emerging knowledge of the past, but in order to improve, you need to develop your knowledge of the past.’ These kind of comments are not very useful as feedback because whilst they may be accurate, they are not helpful. How is a pupil supposed to respond to such feedback? As Dylan Wiliam says, feedback like this is like telling an unsuccessful comedian that they need to be funnier.